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Few-Body Systems

, Volume 54, Issue 11, pp 2113–2124 | Cite as

Bound States of Energy Dependent Singular Potentials

  • R. Yekken
  • M. Lassaut
  • R. J. Lombard
Article

Abstract

We consider attractive power-law potentials depending on energy through their coupling constant. These potentials are proportional to 1/|x| m with m ≥ 1 in the D = 1 dimensional space, to 1/r m with m ≥ 2 in the D = 3 dimensional space. We study the ground state of such potentials. First, we show that all singular attractive potentials with an energy dependent coupling constant are bounded from below, contrarily to the usual case. In D = 1, a bound state of finite energy is found with a kind of universality for the eigenvalue and the eigenfunction, which become independent on m for m > 1. We prove the solution to be unique. A similar situation arises for D = 3 for m > 2, except that, in this case, the solution is not directly comparable to a bound state: the wave function, though square integrable, diverges at the origin.

Keywords

Wave Function Dimensional Space Ground State Energy Attractive Potential Ground State Wave Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  1. 1.Faculté de Physique USTHBBab EzzouarAlgeria
  2. 2.Groupe de Physique ThéoriqueInstitut de Physique NucléaireOrsay CedexFrance

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